chapter 6:the normal distribution & other continuous distributions
TRANSCRIPT
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Copyright 2012 Pearson Education, Inc. publishing as Prentice Hall Chap 6-1 Chap 6-1
Chapter 6
he !or"al #istribution $ %therContinuous #istributions
Basic Business Statistics12thEdition
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&earning %b'ecti(es
In this chapter, you learn: o co"pute probabilities )ro" the nor"al distribution
Ho* to use the nor"al distribution to sol(e business
proble"s o use the nor"al probability plot to deter"ine *hether
a set o) data is appro+i"ately nor"ally distributed
o co"pute probabilities )ro" the uni)or" distribution
o co"pute probabilities )ro" the e+ponentialdistribution
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Continuous Probability #istributions
continuous rando" (ariableis a (ariable thatcan assu"e any (alue on a continuu" canassu"e an uncountable nu"ber o) (alues/
thicness o) an ite" ti"e reuired to co"plete a tas te"perature o) a solution height, in inches
hese can potentially tae on any (aluedepending only on the ability to precisely andaccurately "easure
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he !or"al #istribution
Bell Shaped Symmetrical Mean, Median and Mode
are Equal
Location is determined by themean,
Spread is determined by thestandard de!iation, "
#he random !ariable has anin$inite theoretical ran%e:3 to
Mean& Median& Mode
'
$(')
4
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he !or"al #istribution#ensity unction
2)(X
2
1
e21f(X)
=
he )or"ula )or the nor"al probability density )unction is
Where e = the mathematical constant approximated by 2.71828
= the mathematical constant approximated by .1!1"#
= the pop$lation mean
% = the pop$lation standard de&iation
X = any &al$e of the contin$o$s &ariable
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By !aryin% the parameters and ", *e obtaindi$$erent normal distributions
7any !or"al #istributions
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he !or"al #istribution9hape
'
$(')
"
Changing shi)ts thedistribution le)t or right.
Changing 4increasesor decreases thespread.
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he 9tandardi;ed !or"al
nynor"al distribution *ith any "ean andstandard de(iation co"bination/ can be
trans)or"ed into the standardi;ed nor"aldistribution
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ranslation to the 9tandardi;ed!or"al #istribution
ranslate )ro" = to the standardi;ed nor"althe ?
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he 9tandardi;ed!or"al #istribution
lso no*n as the ?
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Co"paring = and < units
+
.-
/0
./ .'
1ote that the shape o$ the distribution is the same,only the scale has chan%ed0 2e can e3press theproblem in the ori%inal units (' in dollars) or in
standardi4ed units (+)
B 100, 4 50/
B 0, 4 1/
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inding !or"al Probabilities
a b '
$(') P a ' b /5
Probability is "easured by the areaunder the cur(e
5
P a ' b /
!ote that the probability
o) any indi(idual (alue is;ero/
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$(')
'
Probability asrea Fnder the Cur(e
0.50.5
he total area under the cur(e is 1.0, and the cur(e issy""etric, so hal) is abo(e the "ean, hal) is belo*
1.0/=P =
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he 9tandardi;ed !or"al able
he Cu"ulati(e 9tandardi;ed !or"al tablein the te+tboo ppendi+ table E.2/gi(es theprobability less thana desired (alue o) < i.e.,)ro" negati(e in)inity to
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he 9tandardi;ed !or"al able
he (alue *ithin the
table gi(es theprobability )ro" < up to the desired
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eneral Procedure )orinding !or"al Probabilities
#ra* the nor"al cur(e )or the proble" in ter"s o) =
ranslate =-(alues to
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inding !or"al Probabilities
&et = represent the ti"e it taes in seconds/to do*nload an i"age )ile )ro" the internet.
9uppose = is nor"al *ith a "ean o) 1:.0
seconds and a standard de(iation o) 5.0seconds. ind P= G 1:.6/
-06
'-0
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&et = represent the ti"e it taes, in seconds to do*nload an i"age )ile)ro" the internet.
9uppose = is nor"al *ith a "ean o) 1:.0 seconds and a standardde(iation o) 5.0 seconds. ind P= G 1:.6/
+0-/'-06-
B 1:4 5
B & " & -
(continued)
inding !or"al Probabilities
'.12".'
8.'118.*
%
X
=
=
=
P= G 1:.6/ P< G 0.12/
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+
0-/
< .00 .01
0.0 .5000 .500 .50:0
.5>: .5:
0.2 .58> .5:2 .5:81
0. .618> .6218 .6255
9olutionA inding P< G 0.12/
0.58:0/
0- 058:
9tandardi;ed !or"al Probabilityable Portion/
0
P< G 0.12/
P= G 1:.6/
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inding !or"alFpper ail Probabilities
9uppose = is nor"al *ith "ean 1:.0and standard de(iation 5.0.
!o* ind P= J 1:.6/
'
-06
-0
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!o* ind P= J 1:.6/
(continued)
+
0-/
+
0-/
0.58:
1.000 1.0 - 0.58: 0.522
P= J 1:.6/ P< J 0.12/ 1.0 - P< K 0.12/
1.0 - 0.58: 0.522
inding !or"alFpper ail Probabilities
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inding a !or"al ProbabilityLet*een *o Dalues
9uppose = is nor"al *ith "ean 1:.0 andstandard de(iation 5.0. ind P1: G = G 1:.6/
P1: G = G 1:.6/
P0 G < G 0.12/
+0-/'-06-
'"
8118
%
X =
=
=
'.12"
8118.*
%
X =
=
=
Calculate
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+
0-/
9olutionA inding P0 G < G 0.12/
0.08:
0
P0 G < G 0.12/
P1: G = G 1:.6/
P< G 0.12/ M P< K 0/
0.58: - 0.5000 0.08:
0.5000
< .00 .01
0.0 .5000 .500 .50:0
.5>: .5:
0.2 .58> .5:2 .5:81
0. .618> .6218 .6255
0/
0- 058:
9tandardi;ed !or"al Probabilityable Portion/
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9uppose = is nor"al *ith "ean 1:.0and standard de(iation 5.0.
!o* ind P18. G = G 1:/
'
-80;-0
Probabilities in the &o*er ail
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Probabilities in the &o*er ail
!o* ind P18. G = G 1:/
'-80; -0
P18. G = G 1:/
P-0.12 G < G 0/ P< G 0/ M P< K -0.12/
0.5000 - 0.522 0.08:
(continued)
0.08:
0.522
+
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9teps to )ind the = (alue )or a no*nprobabilityA
1. ind the
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inding the = (alue )or aNno*n Probability
E+a"pleA &et = represent the ti"e it taes in seconds/ to
do*nload an i"age )ile )ro" the internet.
9uppose = is nor"al *ith "ean 1:.0 and standardde(iation 5.0
ind = such that 20O o) do*nload ti"es are less than=.
'= -0
0.2000
+=
(continued)
i d th < l )
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ind the 6
0;
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2. Con(ert to = units using the )or"ulaA
inding the = (alue
8.1
'.")8!.'('.18
%X
=
+=
+=
9o 20O o) the (alues )ro" a distribution*ith "ean 1:.0 and standard de(iation5.0 are less than 1.:0
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Fsing E+cel ith he !or"al#istribution
Chap 6-1
>indin% 1ormal 9robabilities
>indin% ' ?i!en @ 9robability
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Fsing 7initab ith he !or"al#istribution
Chap 6-2
>indin% 9('A) *hen ' is normal
*ith a mean o$ 8 and a standard de!iation o$ /
Cumulati!e istribution >unction
!or"al *ith "ean 8 and standard de(iation 2
+ P = G + /
5 0.15:655
1
2
F i 7i it b ith h ! l
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Fsing 7initab ith he !or"al#istribution
Chap 6-
(continued)
1
2
>indin% 3 so that 9('3) & 0- *hen ' is normal
*ith a mean o$ 8 and a standard de!iation o$ /
In!erse Cumulati!e istribution >unction
!or"al *ith "ean 8 and standard de(iation 2
P =G + / +
0.1 .6>0
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E(aluating !or"ality
!ot all continuous distributions are nor"al It is i"portant to e(aluate ho* *ell the data set is
appro+i"ated by a nor"al distribution. !or"ally distributed data should appro+i"ate the
theoretical nor"al distributionA he nor"al distribution is bell shaped sy""etrical/
*here the "ean is eual to the "edian. he e"pirical rule applies to the nor"al distribution.
he interuartile range o) a nor"al distribution is 1.standard de(iations.
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E(aluating !or"ality
Co"paring data characteristics to theoreticalproperties
Construct charts or graphs or s"all- or "oderate-si;ed data sets, construct a ste"-and-lea)
display or a bo+plot to chec )or sy""etry or large data sets, does the histogra" or polygon appear bell-
shapedQ
Co"pute descripti(e su""ary "easures #o the "ean, "edian and "ode ha(e si"ilar (aluesQ Is the interuartile range appro+i"ately 1. 4Q Is the range appro+i"ately 6 4Q
(continued)
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E(aluating !or"ality
Co"paring data characteristics to theoreticalproperties %bser(e the distributiono) the data set
#o appro+i"ately 2R o) the obser(ations lie *ithin "ean S1standard de(iationQ
#o appro+i"ately :0O o) the obser(ations lie *ithin "eanS1.2: standard de(iationsQ
#o appro+i"ately >5O o) the obser(ations lie *ithin "ean S2
standard de(iationsQ E(aluate nor"al probability plot
Is the nor"al probability plot appro+i"ately linear i.e. a straightline/ *ith positi(e slopeQ
(continued)
Constr cting T antile T antile
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Constructing Tuantile-Tuantile!or"al Probability Plot
!or"al probability plot rrange data into ordered array
ind corresponding standardi;ed nor"al uantile(alues
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uantile-uantile nor"alprobability plot )or data )ro" a
nor"al distribution *ill be
appro+i"ately linearA
0
60
>0
-2 -1 0 1 2 Chap 6->
Tuantile-Tuantile !or"alProbability Plot Interpretation
&e)t-9e*ed Uight-9e*ed
Uectangular
0
60
>0
-2 -1 0 1 2 0
-2 -1 0 1 2 0
-2 -1 0 1 2 1O o) the obser(ations are *ithin 1standard de(iation o) the "ean. In a nor"aldistribution this percentage is 6:.26O.
:5.O o) the obser(ations are *ithin 1.2:standard de(iations o) the "ean. In a nor"aldistribution this percentage is :0O./
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E(aluating !or"alityn E+a"pleA Lond unds Ueturns
Chap 6-
(continued)
#escripti(e 9tatistics >6.20O o) the returns are *ithin 2 standard
de(iations o) the "ean. In a nor"aldistribution, >5.O o) the (alues lie *ithin 2standard de(iations o) the "ean./
he se*ness statistic is 0.>0:5 and the
urtosis statistic is 2.56. In a nor"aldistribution each o) these statistics euals ;ero./
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E(aluating !or"alityn E+a"pleA Lond unds Ueturns
(continued)
Plot is not a straightline and sho*s thedistribution is se*edto the right. he
nor"al distributionappears as a straightline./
Tuantile-Tuantile !or"al Probability Plot ro" E+cel
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E(aluating !or"alityn E+a"pleA Lond unds Ueturns
(continued)
Plot is not a straightline, rises uicly inthe beginning, risesslo*ly at the end and
sho*s the distributionis se*ed to theright.
!or"al Probability Plot ro" 7initab
E l ti ! lit
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E(aluating !or"alityn E+a"pleA 7utual unds Ueturns
Conclusions he returns are right-se*ed he returns ha(e "ore (alues *ithin 1 standard
de(iation o) the "ean than e+pected he range is larger than e+pected "ostly due to the
outlier at 2/ !or"al probability plot is not a straight line %(erall, this data set greatly di))ers )ro" the
theoretical properties o) the nor"al distribution
(continued)
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he Fni)or" #istribution
he uni)or" distributionis a probability
distribution that has eual probabilities
)or all possible outco"es o) the rando"(ariable
lso called a rectangular distribution
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Properties o) the
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Properties o) theFni)or" #istribution
he "ean o) a uni)or" distribution is
he standard de(iation is
2
baB
+=
12
a/-b4
2
=
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Fni)or" #istribution E+a"ple
E+a"pleAFni)or" probability distribution o(er the range 2 K = K 6A
2 6
0.25
)=/ 0.25 )or 2 K = K 66 - 21
=
)=/
2
62
2
baB =
+=
+=
158.112
2/-6
12
a/-b4
22
===
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Fni)or" #istribution E+a"ple
E+a"pleAFsing the uni)or" probabilitydistribution to )ind P K = K 5/A
2 6
0.25
P K = K 5/ Lase/Height/ 2/0.25/ 0.5
=
)=/
(continued)
5
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he E+ponential #istribution
%)ten used to "odel the length o) ti"ebet*een t*o occurrenceso) an e(ent theti"e bet*een arri(als/
E+a"plesA i"e bet*een trucs arri(ing at an unloading doc i"e bet*een transactions at an 7 7achine i"e bet*een phone calls to the "ain operator
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he E+ponential #istribution
=Ve1=/ti"eParri(al =unction
E+ponential *ith "ean 0.05
+ P = G + /
0.1 0.:6665
Calculatin% the probability that an e3ponential distribution *ith amean o$ -/ is less than 0-
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Chapter 9u""ary
Presented ey continuous distributions nor"al, uni)or", e+ponential
ound probabilities using )or"ulas and tables
Uecogni;ed *hen to apply di))erent distributions
pplied distributions to decision proble"s
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Dn Line #opic
he !or"al ppro+i"ation o heLino"ial
Basic Business Statistics12thEdition
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&earning %b'ecti(es
In this topic, you learn: hy using a continuity ad'ust"ent yields a "ore
accurate appro+i"ation
o appro+i"ate bino"ial probabilities using the nor"aldistribution
Fsing !or"al #istribution o
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Fsing !or"al #istribution oppro+i"ate Lino"ial Probability
bino"ial distribution is a discrete distribution*hich can only tae on the (alues o) 0, 1, 2, . . ,n.
hen n gets large the calculations associated*ith the bino"ial distribution beco"e tedious.
In these situations can use a nor"al distribution
*ith the sa"e "ean and standard de(iation asthe bino"ial to appro+i"ate the bino"ialprobability
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or a bino"ial rando" (ariable =, P= c/ is non;ero)or c 0, 1, 2, . . . n.
or a nor"al rando" (ariable , P c/ )or any (alue
c is ;ero. 9o to appro+i"ate a bino"ial probability using the
nor"al distribution ha(e to use a continuity ad'ust"ent. I) = is bino"ial and is nor"al *e appro+i"ate P=c/
by Pc M 0.5 G G c 3 0.5/ *here has the sa"e"ean and standard de(iation as =. dding and subtracting the 0.5 is the continuity
ad'ust"ent
he !eed or Continuity d'ust"ent
hen Can he !or"al
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hen Can he !or"alppro+i"ation Le Fsed
he nor"al appro+i"ation can be used as long asA nW X 5 and n1 M W/ X 5
Uecall 7ean o) a bino"ial is B nW
9tandard de(iation o) a bino"ial is 49TUnW1 M W//
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n E+a"ple
You select a rando" sa"ple o) n 1600 tires)ro" a production process *ith a de)ect rate o):O. You *ant to calculate the probability that
150 or )e*er tires *ill be de)ecti(e. Here B 1600Z0.0: 12: and 4
9TU1600Z0.0:Z0.>2/ 10.:5.
&et = be a nor"al rando" (ariable *ith this"ean and standard de(iation then the desiredprobability is P= G 150.5/
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E+a"ple Con[t/
his is P< G150.5 M 12:/R10.:5/ 0.>:0:
9o *e appro+i"ate the probability o) )inding150 or )e*er de)ects as 0.>:0:.
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opic 9u""ary
In this topic, you learned: hy using a continuity ad'ust"ent yields a "ore
accurate appro+i"ation
o appro+i"ate bino"ial probabilities using the nor"aldistribution